Equilateral Triangle

Sum of Areas of Equilateral Triangles Inscribed in Circles

Problem
An equilateral triangle is inscribed within a circle whose diameter is 12 cm. In this triangle a circle is inscribed; and in this circle, another equilateral triangle is inscribed; and so on indefinitely. Find the sum of the areas of all the triangles.
 

09 Dimensions of smaller equilateral triangle inside the circle

Problem
From the figure shown, ABC and DEF are equilateral triangles. Point E is the midpoint of AC and points D and F are on the circle circumscribing ABC. If AB is 12 cm, find DE.
 

Two equilateral triangles inside a circle

 

10 Area common to three squares inside the regular hexagon

Problem
Three squares are drawn so that each will contain a side of regular hexagon as shown in the given figure. If the hexagon has a perimeter of 60 in., compute the area of the region common to the three squares. The required area is the shaded region in the figure.
 

Equilateral triangle bounded by three squares

 

06 Circular arcs inside and tangent to an equilateral triangle

Example 06
The figure shown below is an equilateral triangle of sides 20 cm. Three arcs are drawn inside the triangle. Each arc has center at one vertex and tangent to the opposite side. Find the area of region enclosed by these arcs. The required area is shaded as shown in the figure below.
 

Circular arcs inside a triangle

 

05 Three identical cirular arcs inside a circle

Example 05
Circular arcs of radii 10 cm are described inside a circle of radius 10 cm. The centers of each arc are on the circle and so arranged so that they are equally distant from each other. Find the area enclosed by three arcs shown as shaded regions in the figure.
 

The Regular Tetrahedron

Regular tetrahedron is one of the regular polyhedrons. It is a triangular pyramid whose faces are all equilateral triangles.
 

 
 
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